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Majorant estimates for the percolation threshold of a Bernoulli field on a square lattice

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Abstract

We propose a method for obtaining a monotonically decreasing sequence of upper bounds of the percolation threshold of a Bernoulli random field on ℤ2. On the basis of this sequence, we develop a method for the construction of approximations with guaranteed exactness estimate for the percolation probability. The first term of the sequence is calculated (c 2 = 0.74683).

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References

  1. Yu. P. Virchenko, “Percolation,” in: Mathematical Physics: Big Russian Encyclopedia [in Russian], Moscow (1998), pp. 346–347.

  2. J. M. Hammersley, “Percolation processes. Lower bounds for the critical probability,” Ann. Math. Statist., 28, 790–795 (1957).

    MATH  MathSciNet  Google Scholar 

  3. H. Kesten, Percolation Theory for Mathematicians, Birkhäuser, Boston (1982).

    Google Scholar 

  4. M. V. Men’shikov, S. A. Molchanov, and A. F. Sidorenko, “Percolation theory and some applications,” in: VINITI Series in Probability Theory, Mathematical Statistics, and Theoretical Cybernetics [in Russian], VINITI, Moscow (1986), pp. 53–110.

    Google Scholar 

  5. J. Lamperti, Probability, Benjamin, Amsterdam (1966).

    Google Scholar 

  6. F. R. Gantmakher, Applications of the Theory of Matrices, Wiley, New York (1959).

    Google Scholar 

  7. Yu. P. Virchenko and Yu. A. Tolmacheva, “Revision of the upper estimate of percolation threshold in square lattice,” Mat. Fiz., Analiz, Geom., 10, No. 1, 29–39 (2003).

    MathSciNet  Google Scholar 

  8. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge (1986).

    Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1315–1326, October, 2005.

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Virchenko, Y.P., Tolmacheva, Y.A. Majorant estimates for the percolation threshold of a Bernoulli field on a square lattice. Ukr Math J 57, 1535–1549 (2005). https://doi.org/10.1007/s11253-006-0012-x

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  • DOI: https://doi.org/10.1007/s11253-006-0012-x

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